Charles University in Prague Faculty of Science

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Charles University in Prague Faculty of Science

Photogeneration of charge carriers in substituted polyacetylenes

Fotogenerace nosiˇc˚u n´aboje v substituovan´ych polyacetylenech

Author: Michal Jex Study programme: Chemistry Branch of study: Physical chemistry Supervisor: RNDr. Jiˇr´ı Pfleger, CSc.

Consultants: Mgr. Miroslav Menˇs´ık, Dr.; RNDr. Petr Toman, Ph.D.

Academic year: 2012/2013

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Acknowledgement

I would like to thank my supervisor RNDr. Jiˇr´ı Pfleger, CSc. for suggesting the topic of my thesis, all the help, patience and countless hours of consulta- tions. I would also like to thank my consultants Mgr. Miroslav Menˇs´ık, Dr.

and RNDr. Petr Toman, Ph.D. for all the help and ideas how to improve my thesis.

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Prohl´aˇsen´ı

Prohlaˇsuji, ˇze jsem tuto závˇereˇcnou práci zpracoval samostatnˇe a ˇze jsem uvedl vˇsechny pouˇzité informaˇcn´ı zdroje a literaturu. Tato práce ani jej´ı pod- statná ˇcást nebyla pˇredloˇzena k z´ıskán´ı jiného nebo stejného akademického titulu.

V Praze dne 20.kvˇetna 2013

Michal Jex

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Abstract

We present an improved model of charge carrier photogeneration inπ-conjugated polymers with weak intermolecular interactions based on the model of Arkhipov. It includes quantum effects affecting the creation of charge transfer states, which occurs as an intermediate step in the free charge carrier photogeneration process. The electrostatic potential between the electron and the hole and transfer integrals needed for the calculation of the potential barrier for the charge transfer state dissociation are calculated quantum-chemically.

We apply our model on experimental data of the charge carrier photogeneration efficiency in poly[1-trimethylsilylphenyl,2-phenyl]acetylene to explain its dependence on applied electric field. We eliminate several problems of the previous model. We are able to fit experimental data with just one set of parameters in the whole interval of the applied electric field. We do not have to consider several intervals of the electric field separately as in the previous work and reduce the number of needed parameters to three.

Key words

π-conjugated polymers, charge carrier photogeneration, photoconductivity Abstrakt

Analyzujeme námi navrˇzený model fotogenerace volných nosiˇc˚u náboje v π-konjugovaných polymern´ıch materiálech se slabými mezimolekulárn´ımi in- terakcemi vycházej´ıc´ı z Archipovova modelu. Zahrnuli jsme kvantové efekty ovlivˇnuj´ıc´ı vznik stav˚u spojených s pˇrenosem náboje pˇredstavuj´ıc´ı mezikrok pˇri fotogeneraci volných nosiˇc˚u náboje. Elektrostatická interakce mezi elekt- ronem a d´ırou, stejnˇe tak jako pˇrenosové integrály, jsou spoˇcteny kvantovˇe- chemicky. Model jsme aplikovali na experimentáln´ı data závislosti fotogenerace volných nosiˇc˚u náboje v poly[1-trimethylsilylfenyl,2-fenyl]acetylenu na intenzitˇe vnˇejˇs´ıho elektrického pole. Podaˇrilo se odstranit nˇekteré nedostatky pˇredchoz´ıho modelu. Byli jsme schopni namodelovat experimentáln´ı data bez nutnosti rozdˇelen´ı intervalu intenzit elektrických pol´ı na nˇekolik oblast´ı, ve kterých bylo nutno prokládat experimentaln´ı data modelovými kˇrivkami s r˚uznými fyzikaln´ımi parametry. Náˇs model je schopen popsat experimentáln´ı data v celém rozsahu intenzit elektrického pole pomoc´ı jedné sady parametr˚u modelu. Dále se nám podaˇrilo sn´ıˇzit poˇcet parametr˚u modelu na tˇri.

Kl´ıˇcov´a slova

π-konjugovan´e polymery, fotogenerace nosiˇc˚u n´aboje, fotovodivost

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Chapter 1 Introduction

In this work we study photogeneration of free charge carriers in poly[1- trimethylsilylphenyl,2-phenyl]acetylene, which is an example of one dimen- sional organic semiconductor. Poly[1-trimethylsilylphenyl,2-phenyl]acetylene is an example of a polymer with well separated π-conjugated system on the polymer backbone. We further generalize Arkhipov model first presented in [1] and revisited in [2, 3]. We take into consideration several quantum effects which were either neglected in the previous models or approximated using classical methods.

The reason for studying the photogeneration of free charge carriers is the following: it is a principal process in photovoltaic cells, photoelectric sen- sors and xerographic photosensitive layers. The photogeneration as well as the transport of free charge carriers follows different mechanism as compared to inorganic semiconductors. Unlike the inorganic compounds, we are not able to describe this process within the energy band structure of the material. The reason for this is a weaker interaction between the molecules in organic compounds which leads to energy bands too narrow to be able to explain the photogeneration of free charge carriers and the conductivity of the material. Another difference might be observed in dielectric constant: in inorganic semiconductors its value usually exceeds 10 but in organic materials its value is around 3. Due to the smaller values of the dielectric constant in organic materials the mutual Coulomb attraction force in an electron-hole pair is much stronger. After the photoexcitation of the organic semiconduc- tive materials an exciton is created as a primary quasiparticle instead of a free electron and a hole. The exciton is essentially the pair of electron and

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hole bounded mutually by Coulomb attraction. After the creation of the electron-hole pair further activation energy is needed for the creation of free charge carriers. This process was previously a topic for a number of scientific investigations.

In Chapter 2 we summarize several previously developed models used for the description of the photogeneration of free charge carries in organic materials along with the essential basics of ab initio chemical calculations.

Chapter 3 is focused on molecular structure of our polymer. By means of density functional theory we calculated the structure and the geometry of the decamer model of our polymer. We present an optimization of the geometry of the neutral molecule and ion-radicals. We show a basic approach of calculating the position of the cation-radical and the anion-radical with respect to each other. Spin densities and charge distribution were calculated, too.

In Chapter 4 we present our improvements of the Arkhipov model of photogeneration of free charge carriers. The interaction between the anion- radical and a test charge was calculated. We showed the calculated transfer integral between the neighbor monomer units dependence on the dihedral angle between them. A detail analysis of the calculation of the potential barrier is given.

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Chapter 2 Theoretical background

In this chapter we summarize existing models of photogeneration of free charge carriers in organic compounds. We introduce a modified Arkhipov model, which we improve later on. Some essential basics connected with ab initio quantum calculation are given at the end of this chapter.

2.1 Photogeneration of free charge carriers in organic compounds

First models explaining the formation of free charge carriers upon photoexcitation were developed to describe photoconductivity in organic molecular crystals. According to [4] the photogeneration of free charge carriers in organic compounds is connected, first, with the formation of an exciton after photoexcitation and, in the second step, with a subsequent dissociation of this exciton into free charge carriers. The exciton can be described as a bound state of the electron and the hole attracted to each other by the Coulomb force. The dissociation of the exciton is a thermally activated process. In conjugated polymers the concept of intrachain and interchain exciton is fre- quently used. The interchain exciton is usually called as a CT state, which can be described as a bounded electron-hole pair, where the hole and the electron are located on different molecules. Excitons can be either mobile or localized on an energy level below the edge of the conduction band.

The description of photogeneration of free charge carriers has been treated by many models. These models differ in the way how the bound pair of the

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hole and the electron is formed and in the way how these pairs dissociate.

The problem concerning the formation of bounded charge couples has not been fully solved yet. Bounded charge pairs can be formed either by direct photoexcitation or indirectly. Direct transitions are observed on molecular crystals unlike the polymers where only a indirect absorbtion is observed.

Indirect formation of CT states starts with photoexcitation of the molecule to a higher singlet state followed by one of the following transitions:

a) autoionization of the excited state and thermalization of the localized hole and of the hot electron leading to an electrostatically bound pair

b) electron jumps to a neighboring molecule after the relaxation into the first excited singlet state

c) transformation of excitation energy from a donor to an acceptor and creation of the CT complex in the donor-acceptor pair.

The efficiency of the photogeneration of free charge carriers is described by a product of CT state creation probability and CT state dissociation probability. The efficiency of the CT state creation is described by a primary quantum yield and is denoted by η0. The primary quantum yield describes probability of the CT state creation after absorption of one photon. This process is usually considered as independent of temperature and external electric field and dependent only on the energy of a photon. How- ever, generally this process can depend on external electric field and phonons.

The primary quantum yield is also affected by competing transitions in the molecule. Usually, before the creation of the CT state, the exciton undergoes migration, which can be up to tens nanometers in length. Such a distance is usually much smaller than the penetration depth of the absorbed light in organic compounds. This lowers the efficiency of organic solar cells and introduces a necessity of creating special nanostructures to increase the efficiency of photogeneration of free charge carriers.

Two basic theoretical models of the CT state dissociation can be found in the literature. The first one is a so-called Poole-Frenkel model. It is based on lowering Coulomb barrier between the hole and the electron in an external electric field. This model does not take into consideration a diffusion process. The second model is more sophisticated and takes into consideration drift and diffusive motion of the charge carrier in the internal and external electric field. This approach can be found in both the Onsager and Noolandi- Hong theories.

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In the following subsections we will present some older models of photogeneration. A more complete description of these models can be found in [4].

2.1.1 Ballistic model

The ballistic model was developed by Silinsh[4, 5]. It was first used to describe photogeneration of free charge carriers in molecular crystals. This model considers photogeneration of free charge carriers as a process consisting of several steps. First step is a photoexcitation of the molecule followed by autoionization of the excited state. This results in a localized hole and a hot quasi-free electron, but depending on the organic material the electron can be localized and the hole can be mobile. Thermalization of the hot particle ends with the bounded electron-hole pair in which the electron and the hole are separated by a distance r₀. This model assumes that the dependence of the thermalization distance r₀ between charges is proportional to the square root of the surplus energy of the electron E_k. The energy E_k increases with the energy of the absorbed photon. These two physical quantities can be expressed as

r₀ =p

Dτ_th=

sDE_k

hν_ph² (2.1)

where τth is the thermalization time,D is the diffusion coefficient, νph is the phonon frequency and h is the Planck constant. The autoionization yield is given by

η0(hν) = k_AI(hν) k_AI(hν) +P

ik_i(hν) (2.2)

wherek_AI and k_i are the rate constants of autoionization and intramolecular loss channels. It is assumed that η₀ can be larger for higher excited states.

The primary quantum yieldη₀ can generally depend on the absorbed photon energy.

The hot electron is assumed to be a Brownian particle with excess kinetic energy. This energy is dissipated by non-elastic scattering and acts against the Coulomb interaction between the hole and the hot electron. Due to the diffusive motion of the electron the thermalization distancer₀ is subjected to

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a dispersion. The distribution of the thermalization distance can be described by the Gaussian distribution as follows

p(r) = 1

4π³²r²₀σexp

−(r−r₀)² σ²

(2.3) where σ is a parameter describing dispersion. The thermalization distance in solids is not higher than 10−15 nm and σ² hr₀i². For this situation Gaussian distribution function can be approximated by properly normalized δ function

p(r) = δ(r−r0)

4πr²₀ (2.4)

with only one parameter.

2.1.2 Onsager model

The Onsager model is popular and widely used for interpretation of photogeneration in polymers and molecular crystals [4]. The creation of the electron-hole pair is described in the same way as in the ballistic model, i.e.

as an indirect process. The difference between these two models arises in the modeling of the dissociation of the hole-electron pair with initial separation distancer₀. This problem is treated within the Onsager diffusion theory.

In this model the primary quantum yieldη₀ is assumed to be independent on temperature T and external electric field F; all these dependencies are included in the dissociation step. The quantum yield of photogeneration of free charge carriers can be described as

η(F) =η₀

π/2

Z

−π/2 2π

Z

0

∞

Z

0

g(r, θ)f(F, r, θ)r²sin(θ)dθdφdr (2.5)

where g(r, θ) is the initial space distribution of thermalized charge pairs, r is the distance between the hole and the electron, θ is the angle between the direction of the external electric field and the vector connecting the hole and the electron. The function f(F, r, θ) describes the probability that the hole- electron pair does not recombine. The functionf(F, r, θ) fulfills the following boundary conditions f(F,0, θ) = 0 and f(F,∞, θ) = 1. The dissociation of

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the bound pair is described as Brownian motion of a particle in the Coulomb field of electron-hole pair and the external electric field F. The electric potential energy U can be written as

U =−eF rcosθ− e²

4π₀_rr (2.6)

whereF cosθ is the projection of the external electric field into the direction of the vector connecting the electron-hole pair, ₀ is the permittivity of the vacuum, _r is the relative permittivity of the material, e is the elementary charge and r is the distance between the hole and electron. The function f(F, r, θ) can be calculated from the time independent Smoluchowski diffusion equation

div(exp(−U/(kT)∇f)) = 0. (2.7)

The distribution of thermalized pairs g(r, θ) is described by δ function in solids and by Gaussian and exponential distribution functions in liquids.

Separation distances obtained in this model by fitting experimental data are between 2−3 nm [4]. Such high separation distances are hard to explain theoretically.

2.1.3 Knights-Davis model

This model, similarly as the first one, considers indirect creation of the bound pair [6]. It was developed to describe experimental results of photogeneration of free charge carriers in amorphous selenium. In this model we consider photogeneration of free charge carriers as a two-step process. The first step is the absorption of a photon followed by a thermalization leading to electron- hole bound pair. The thermalization of the electron-hole pair is accompanied by diffusion of the electron similarly as in the ballistic model. The second part of the photogeneration of free charge carriers is described by Poole-Frenkel mechanism as a dissociation of the bound pair facilitated by an external electric field. Quantum yield of the photogeneration of free charge carriers is determined by the ratio of the recombination time τr and ionization timeτi

as

η = 1

1 +τrτ_i⁻¹. (2.8)

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Ionization time can be calculated in the following way

τ_i =ν_e⁻¹exp

e²

4πr0 +eF r₀−q

e³F π

kT (2.9)

where νe is the frequency factor and is the permittivity of the material.

2.1.4 Noolandi-Hong model

This model was developed for the description of the photogeneration of free charge carriers in phthalocyanine thin films [7]. Experimental data on phthalocyanine showed the quantum yield independent on the excitation energy.

This is explained in the following way; in the first step the molecule, excited by absorption of a photon to the higher excited state, looses surplus energy by fast internal conversion and the molecule stays in the first excited state with a relatively long lifetime. In the second step the excited state can either relax to the ground state or it can be autoionized forming a bound charge pair. The hole and the electron in the bound pair are located at different neighboring molecules. The probability of creation of the bound pair can be written as

k(θ) =k₀exp(F F₀⁻¹cos(θ)) (2.10) where k₀, F₀ are constants and θ is the angle between the vector of the external electric field F and the vector connecting the electron and the hole.

The probability of dissociation of the bound pair is obtained similarly as in the Onsager model. This probability can be obtained as a solution of the time-independent Smoluchowski equation (2.7). The boundary conditions in this model are different from those in the Onsager model. The Noolandi- Hong model considers also a recombination of bound pairs on a sphere of diameter rm with finite recombination rate K, where rm is a mean lowest intramolecular distance. The independence of the quantum yield on the energy of absorbed photons is caused by the losses of the surplus energy of the bound pair. The losses of the energy are caused by the internal conversion in the material.

Quantum yield for this model can be expressed as η(F) = k(F)

k(F) +k_n+k_r (2.11)

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where k_r and k_n are rate constants of the radiative and non-radiative transitions of the excited state to the ground state and k(F) is

k(F) = 1 2

1

Z

−1

k(cosθ)Ω(F, r,cos(θ))d(cosθ) (2.12)

where Ω is the dissociation probability of bound pair separated by a distance r. This model has three tunable parameters: F₀, ^kⁿ_k^+k^r

0 and ^K_D. One special limit of these parameters gives us the Onsager model.

2.2 Arkhipov model

In this section we present the Arkhipov model described in [1]. This model was designed to explain the photogeneration of free charge carriers in conjugated polymers doped by either electron donors or acceptors. Similarly as other models it consists of two steps. In the first step an exciton transforms into a coulombically bound pair. This step is realized in the place of a charge transfer, which is usually localized on the site of a dopant or alternatively in the neighborhood of a deep potential well. In the second step the bound pair dissociates into free charge carriers in an applied external electric field. The description of temperature and electric field dependencies of exciton transformation into Coulomb bound pair is not fully explored within this model.

It is assumed that after the absorption of a photon the created exciton travels along the polymer backbone. It moves through several conjugated segments before it either relaxes into the ground state or transforms into the bound state in the place of charge transfer. The probability of the exciton dissociation w_dcan be calculated from the relative rate of exciton relaxation without dopants and from the probability of the tunneling of the electron to the electron acceptor in the distance d from the main polymer backbone.

These processes can be described using the mean exciton lifetime τ and the tunneling rate ν₀exp(−2γd), where ν₀ is the frequency factor and γ is the inverse localization length. Putting it together we obtain the total probability of the exciton dissociation in the following form

wd= 1

1 + ^exp(2γd)_ν _τ (2.13)

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It can be easily seen that the exciton dissociation rate decays exponentially with growing d.

After the exciton dissociation the electron is localized in a distance d from the polymer backbone. The hole remaining on the polymer backbone is trapped in a potential well U(r), which is obtained as a sum of the external electric field F and Coulomb electric field of the localized electron. U(r) can be written in the following form

U(r) = −eF zr− e² 4π0r

p(r² +d²) (2.14) where z = cos(θ), θ is an angle between the direction of the external electric field and the polymer backbone, r is the distance measured along the polymer backbone and F is the absolute value of the external electric field.

For sufficiently large F or d the function U(r) monotonously decreases with respect to r. When such situation occurs, the charge carriers become free immediately after the exciton dissociation. If this is not the case the function U(r) has a local minimum for certain distance r = r_min around which the hole is trapped. We assume that the hole can escape the potential well by thermal activation. In order to calculate the escape probability of the hole, we need to calculate the potential barrier. This is done by approximating the potential well as a harmonic potential. By the help of the Taylor expansion to the second order we rewrite U(r) aroundr_min and we obtain the potential U_osc(r) in the form

Uosc(r) =−eF zrmin− e² 4π0r

pr_min² +d² + e²(d²−2r²_min)

8π₀_r(r²_min+d²)⁵²(r−rmin)² (2.15) The last part of the previous equation describes the energy of the oscillating hole within the potential well. The minimal energy of the charge carrierE_min can be calculated as a sum of the minimum of the potential U_osc(r) and the energy of zero point oscillation. We obtain the following

E_min =U_osc(r_min) + 1

2~ω =−eF zr_min− e²

4π₀_r(r_min² +d²)¹² +~

s e²(d²−2r_min² ) 16π₀_r(r_min² +d²)⁵²m_{ef f}

(2.16)

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where ~ is reduced Planck constant, ω is the oscillation frequency and m_{ef f} is the effective mass of the charge carrier. The equation (2.16) assumes an infinitely long polymer chain. In a real system the hole can move freely only within a conjugated segment. This narrows the potential well and affects oscillation energy of the hole.

The height of the potential barrier needed to be overcome by the hole can be calculated as a difference between the local maximum of the potential Umax =U(rmax) and the minimal energy Emin. There are two limiting cases when the dissociation of the bounded state happens immediately after the exciton dissociation. One of them was already discussed above, it is when the potentialU(r) is monotonous with respect to r. Another one is when there is a potential well, but it is so shallow that the harmonic approximation done in (2.15) is unacceptable and, therefore, the minimal energy of the hole would be formally greater than the local maximum of the potential Emin > Umax. It can be shown by direct calculation that the height of the potential well increases with decreasing d. Because the potential barrier depends on z we can find such a functiond0(z) for whichEmin(d0, z) = Umax(d0, z) is satisfied.

It means that we can find a critical initial separation distance depending on z for fixed F. For values d greater then this critical value, i.e. for such d which fulfills

d > d₀(z), (2.17) the dissociation of CT state occurs immediately after the exciton dissociation.

This is in reality not very probable because with increasingd the probability of CT state creation decreases exponentially. When the condition (2.17) is not fulfilled additional thermal activation is needed to create free charge carriers. The hole has to escape from the potential well before recombination with the trapped electron. We denote the rate of the hole recombination as νrec. The recombination rate is governed by the tunneling rate of the electron and it can be written as

ν_rec =ν₀exp(−2γd) (2.18)

The escape of the hole from the potential well is a thermally activated process and its rate can be written as

νesc =ν0exp

E_min(d)−U_max(d) kT

(2.19)

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Now, if we combine equations (2.13) and (2.18) we obtain the total probability of creation of free charge carrier from one photon absorption w=w(d, z) depending on d and z as

w= [1 + (ν₀τ)⁻¹exp(2γd)]⁻¹

1 + exp[−2γr+U_max(d, z)−E_min(d, z)

kT ]

−1

(2.20) the expression (2.20) describes photogeneration of free charge carriers for fixeddand θ. In a real polymer material molecules are randomly oriented as well as electron acceptors are randomly distributed. If we are interested in the quantum yield of the photogeneration of free charge carriers in such materials we have to average the probability w with respect to the distribution of the distance d of the traps from the polymer chain and also with respect to the random orientation of the polymer chain with respect to the direction of the applied electric field. We assume that distribution of electron acceptors P(d) in a polymer material can be approximated by the Poisson distribution which can be written as

P(d) = 2πrlN_dexp[−πlN_d(d²−d²_min)] (2.21) wherelis the length of a conjugated segment,d_min is minimal distance of the electron acceptors from the polymer backbone andN_dis the concentration of electron acceptors . The quantum yield η of photogeneration of free charge carriers can be written as

η= 2π Nd

1

Z

0

dz

∞

Z

max{d_min,d0(z)}

dx xexp[^−π_N

d(x²−d²_min)]

1 + (ν0τ)⁻¹exp(2γx)+

+

1

Z

0

dz

max{dmin,d0(z)}

Z

dmin

dx xexp[^−π_N

d(x²−d²_min)]

1 + (ν₀τ)⁻¹exp(2γx)·

·

1 + exp

−2γx+U_max(d, z)−E_min(d, z) kT

−1

.

(2.22)

where we averaged the probability w over the distribution P(d) over the random orientation of the polymer chains.

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2.3 Modified Arkhipov model

A modification of the original model of Arkhipov [1] was done in [2]. The difference between the original model and the modified one is the following; there is a possibility that the charges in CT state can recombinate back forming an exciton. This enables additional cycling in the model. Another difference of this model to the original one is that it deals with a pure polymer, containing neither donors nor acceptors. Instead of an additive, another molecule of the polymer serves as the acceptor of the electron in the photogeneration process.

In this model, similarly as in the original one, we assume that after absorption of a photon the created exciton travels along the π-conjugated system. It can be transformed into a CT state with a probability η₀. The transformation of the exciton into a CT state can occur at points where two polymer chains come close to each other. The transformation of the exciton into the CT state proceeds via an intermolecular jump of the electron. After the jump the hole and the electron could theoretically move independently in an external electric field as long as the Coulomb interaction between them is overcome. In conjugated systems the mobility of the hole µ_d is higher than the mobility of the electron µe and, as a result, holes are the dominant free charge carriers formed after the dissociation process. In this model the hole escapes from the potential well formed around the electron, which is less mobile than the hole. In this aspect it is very similar to the Arkhipov model [1].

The hole is influenced by the potential described in the same way as in the original Arkhipov model, i.e. a sum of potentials of the external electric field and of the Coulomb interaction between the hole and the electron. This potential U(r) can be written as in (2.14) where we denoted as a separation distance between the neighboring chains andras the distance traveled by the hole along the chain away from the localized electron. As a simplification we assume that _r is independent on r. Because we are working with the same potential as in the previous case its properties remain unchanged. We denote the coordinate for which the minimum of the potential U(r) is achieved as r_min. The procedure how to calculate the value of r_min are presented in [1]. The conditions under which this local minimum can be calculated are described in [2]. The escape of the mobile hole trapped in the potential

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well of U(r) is a thermally activated process with a rate constant k_esc. This constant can be calculated in a similar manner as before

k_esc =k₀exp

−(U(resc)−Emin) kT

(2.23) where k₀ is a constant. There are two versions of this model depending on whether we allow the resonant coupling of the exciton state and the CT state or not.

2.3.1 “One-step” model of exciton dissociation and re- combination of CT state

This model coincides with the original Arkhipov model. We assume that the CT state can either dissociate into free charge carriers or recombinate into the ground state and we neglect the possibility of the CT state transformation back into the exciton. We denote the rate constant of the recombination as k_rec. The probability of the CT state dissociation p_esc can be written as

p_esc = k_esc

k_esc+k_rec = 1 1 + ^k_k^rec

0 exp{^U(r^esc_kT^)−E^min}. (2.24) The ratio

A= krec

k₀ (2.25)

can be interpreted as a one-step relative recombination constant. Quantum yield of the photogeneration of free charge carriers can be written as a product of the probability of the CT state creation represented by the primary quantum yield and the probability of the CT state dissociation into free charge carriers. This can be expressed as

η=η0pesc (2.26)

where η0 denotes the primary quantum yield.

2.3.2 Model with resonant coupling of exciton and CT state

In this subsection we discuss the situation when the exciton and the CT state are energetically close to each other or even at a resonance, which

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results in an exciton localization by a self-trapping. Such coupling of the CT state and the exciton leads to multiple transitions between the exciton and the CT state. When such situation occurs, it is impossible to introduce the primary quantum yield as in the previous case and we have to consider effects resulting from “cycling”. In order to do so properly, we need to describe elementary processes in our model and then we have to solve the kinetic equations describing the system. The processes considered after the initial creation of the exciton by photoexcitation in our model are the following:

a) the exciton can either transform into the CT state with a rate constant k₁ or relax into the ground state with a rate constant k_g^EX.

b) the CT state can either dissociate into free charge carriers with a rate constant kesc or relax into the ground state with rate constant k^CT_g or it can be transformed back into the exciton with a rate constant k−1.

All the listed processes are summarized in Figure 2.1.

Figure 2.1: Processes of the photogeneration of free charge carriers with included resonant coupling with their respective rate constants; CT state and exciton coupling (k₁, k−1), loss channels (k_g^EX,k_g^CT), photogeneration of exited state (k_E) a creation of free charge carriers (k_esc).

The quantum yield of the photogeneration of free charge carriersηcan be expressed as the ratio of the concentration of the created free charge carriers and the absorbed photons per unit volume.

η= k_esc[CT] Φabs

(2.27) where Φ_abs denotes the rate of photon absorption per unit volume and [CT] is the concentration of CT states. If we want to calculate the concentration of CT states we need to solve a set of kinetic equations describing the processes

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in Figure 2.1. These equations can be written as d

dt[EX] =fΦ_abs−k₁[EX]−k^EX_g [EX] +k−1[CT] d

dt[CT] =k1[EX]−kesc[CT]−k−1[CT]−k^CT_g [CT],

(2.28)

where f is a rate constant describing the initial singlet state transformation S₁ into exciton and [EX] is the concentration of the excited states. One of the possible ways how to solve equations (2.28) is to use the steady state assumption, i.e. _dt^d[EX] = 0 and _dt^d[CT] = 0. A straightforward results in

η= k₁

k₁+k^EX_g f k_esc

k_esc+k−1+k_g^CT −_k^k⁻¹^k¹

1+k^EX_g

. (2.29)

We can rewrite the previous equation as η=η_{ef f} k_esc

k_esc+k^{ef f}rec

, (2.30)

whereη_{ef f} is the effective primary quantum yield of charge carrier photogeneration

η_{ef f} = k₁

k₁+k^EX_g f₀ (2.31)

and k^{ef f}_rec is the effective rate constant of CT state recombination k^{ef f}_rec =k−1+k_g^CT − k₋₁k₁

k₁+k_g^EX =k^CT_g + k−1k^EX_g

k₁+k_g^EX. (2.32) Formally the equation (2.30), derived to the modified Arkhipov model, and (2.24) belonging to the original Arkhipov model look to be the same, but they differ in the possible dependence of parameters η^{ef f}₀ a k_rec^{ef f}. Contrary to the original model, the modified model parameters are allowed to depend on the external electric field. It effects the fact that external electric field affects energy difference between CT states and, hence, it influences the detailed balance conditions between the occupation of the excited states and CT states. Consequently, the rate constants k₁ and k−1 become dependent on the external electric field and, in the same way, also the rates η^{ef f}₀ a k_rec^{ef f}

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become dependent on the external electric field. The total quantum yield within the modified Arkhipov model can be written as

η=η_{ef f} 1

1 +A_{ef f}exp(^U(r^esc_kT^)−E^min) (2.33) where A_ef f = ^k^{ef f}^rec

k^{ef f}₀ is effective preexponential factor.

2.4 Quantum-chemical calculations

In this section we briefly mention essentials of the ab initio quantum calculations. The task to be solved is to find eigenvectors and eigenvalues of the Hamiltonian in the following form

H =−X

j∈N

~²

2M_j∆_N,j −X

j∈M

~²

2m_e∆_j +X

j∈N

X

k∈N

e²Z_jZ_k 4π₀|Q_j−Q_k|

−X

j∈N

X

k∈M

e²Z_j

4π₀|Q_j−q_k|+X

j∈M

X

k∈M

e² 4π₀|q_j−q_k|

(2.34)

where N and M denotes the number of atoms and electrons in the system respectively, M_j is the mass of the j-th atom nucleus, m_e is the mass of the electron, Z_j is the charge of the j-th atom nucleus, −∆_N,j and −∆_j denotes the Laplace operator ofj-th atom andj-th electron respectively, Qj

and q_k are the positions of thej-th atom and k-th electron respectively and lastly ~is reduced Planck constant. By neglecting the relativistic effects we employed the first simplification. These effects can be added by the means of the perturbation theory or with help of relativistic pseudopotentials [8].

Such system with the Hamiltonian H is not solvable analytically for the case that N +M >2. For this reason we have to employ certain simplifications.

The most important is a Born-Oppenheimer approximation, based on the separation of electrons and atom nuclei. The motion of the nuclei can be then solved either classically or by quantum mechanical calculations. There are many ways how one can address the problem of the electrons. We will briefly describe two methods, which we use in our work.

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2.4.1 Hartree-Fock method

First method, we describe, is so called Hartree-Fock method (HF). We need to solve the equation

H_elψ =Eψ (2.35)

where the operator H_el can be written as H_el=−X

j∈N

~²

2M_j∆_N,j −X

j∈M

~² 2m_e∆_j

−X

j∈N

X

k∈M

e²Z_j

4π₀|Q_j−q_k|+X

j∈M

X

k∈M

e² 4π₀|q_j−q_k|

(2.36)

This equation can be solved exactly for N ≤2. For more than two electrons we have to employ certain approximations. The HF method is a variational method using the approximation of independent electrons [8]. In this method we are searching for the solution of the associated variational problem in the form of a single Slater determinant. The model of independent electrons assumes that the electron moves in the mean electric field generated by all the other electrons in the system. This leads to the self consisting field problem which is solved iteratively. The main problem of this approach is that we are not able to obtain electron correlation correctly. This error can be fixed by many post HF methods, for example by Møller-Plesset perturbation theory, where the correlations are added as a perturbation term. One of the important applications of the HF method is so called Koopman’s Theorem. This theorem states that the ionization potential of a closed-shell molecule can be calculated as a minus energy of the highest occupied molecular orbital. This approach gives surprisingly good results which is caused by the compensation of the multiple errors. For more details we refer the reader to the monograph [8] or [9].

2.4.2 Density functional theory

Another approach we employ later is the density functional theory (DFT).

This method is based on the Hohenberg-Kohn theorems [9]. These theorems state that for the electron system with the non-degenerate ground state without magnetic field the ground state eigenfunction can be expressed as the function of the total electron density. By this approach we are able to

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solve the problem of the function depending on three spatial coordinates only and not on the 3N electron spatial coordinates. The energy of the system is then given by appropriate functional. The problematic part of this process is to find such functionals which would give us the appropriate solution. The most problematic task is to calculate the exchange correlation energy. There is a large variety of different functionals available, from those calculated theoretically to those, which are obtained by fitting the experimental data. For our purpose we chose hybrid functional B3LYP presented in [10, 11]. The hybrid functionals are special case of DFT method which treats the problem with the exchange correlation energy in such a way that the exchange correlation energy is calculated partially from the exact exchange correlation energy from HF method and partially from other sources. The DFT methods are often used for the calculation of oligomers [12, 13]. They are also used for the calculation of various σ-conjugated and π-conjugated systems [14, 15].

Calculations of ion-radical systems can be done within the DFT methods successfully [14, 15].

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Chapter 3 Calculations of the polymer structure

In this chapter we present results of the quantum-chemical calculations of the configuration and of the conformation of the polymer on which the modeling of the charge carrier photogeneration will be performed. Finding the right structure of the polymer chain, mutual position of adjacent chain, as well as the charge distribution in the cation-radical and anion-radical, are essential for later calculations of the photogeneration process. Ab initio quantum-chemical calculations were done using Gaussian software [16].

Chains of the studied polymer were modeled as oligomers, each one con- sisted of 10 monomer units. Poly[1-(trimethylsilyl)phenyl,2-phenylacetylene]

was the polymer under study in this work.

3.1 Geometry of a single polymer chain

The first thing we had to know about the polymer was the configuration of the polymer backbone. We studied theoretically three options how the monomer units can be connected. The studied options were: head-to-head, head-to-tail and tail-to-tail. In Figure 3.1 these configurations are denoted by the type of bonding of the first two monomer units. We considered only the regular configurations. We decided which configuration is the most stable one by comparing the ground state energies of the neutral molecules in its optimal geometries. The energy of those configurations were calculated in the following way. First, we calculated the optimal geometry for our decamers using

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Figure 3.1: Studied possibilities of bonding of monomer units.

the Hartree-Fock method in the basis of atomic orbitals 6-31G* [17, 18]. We took these structures as an initial guess of the geometry for the density functional theory calculation. As a density functional theory functional we chose a hybrid functional B3LYP [10]. In addition to the standard B3LYP functional we have considered also this functional supplemented by an empirical correction term [19] describing the dispersion interaction between the atoms in molecule. The reason for this approach is a relatively strong interaction coming from the Van der Waals forces between the adjacent phenyl groups.

These interactions are not described well in the original B3LYP functional.

The calculations without added dispersion are denoted by abbreviation DFT and the calculations with added dispersion by DFTD. Both calculations by DFT and DFTD method were done in the basis of atomic orbitals 6-31G*.

The calculations have shown that the most stable configuration is the one with the head-to-tail arrangement. However, the difference between the energies of the ground states of these configurations is so small that all three configurations are possible and the final configuration of the polymer could be influenced by the method of the preparation. The ground state energy is summarized in Table 3.1.

Table 3.1: Ground state energy of decamer in Hartrees Configuration B3LYP B3LYP+dispersion

Tail to tail -9482.92165887 -9483.72267259 Head to tail -9482.92251630 -9483.74275046 Head to head -9482.92013676 -9483.72502132

Table 3.2 shows these energies listed in kJ per mole. For the conversion

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we used units the conversion factor 1 Hartree = 2625.4996_mole^kJ [20].

Table 3.2: Ground state energy of decamer in kJ per mole Configuration B3LYP B3LYP+dispersion

Tail to tail -24897407 -24899510 Head to tail -24897409 -24899563 Head to head -24897403 -24899516

The difference between the energy of different configurations was bigger if DFTD method was used. The reason for this is that the dispersion is dependent on the mutual position of phenyl groups. Dispersion is also affected by position of trimethylsilyl groups with respect to each other. In the tail- to-tail configuration and in the head-to-head configuration the trimethylsilyl groups are not so close to each other than they are in the head-to-tail configuration. Experimental study of the configuration of our polymer was not done.

Configurations of all polymer chains look quite similar. The polymer backbone is twisted into helical structure and the side groups of the polymer (phenyl and 4-(trimethylsilyl)phenyl groups) are directed away from each other as much as they are allowed.

It is worth mentioning that for a polymer of infinite length the configuration head-to-head is the same as tail-to-tail. The difference between them arises from the different end groups, where for one configuration there are phenyl side groups and for another one 4-(trimethylsilyl)phenyl side groups.

Visualization of the optimal geometries of different configurations can be seen in Figures 3.2-3.4. Here we omitted hydrogens for better clarity of figures. Silicon atoms are denoted by pink color, carbon atoms of the main backbone are orange and carbon atoms of the sidegroups are grey.

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Figure 3.2: Optimal geometry of the decamer in the tail-to-tail configuration calculated by the DFTD method.

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Figure 3.3: Optimal geometry of the decamer in head-to-tail configuration calculated by the DFTD method.

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Figure 3.4: Optimal geometry of the decamer in head-to-head configuration calculated by the DFTD method.

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In the following graphs we show some basic properties of the polymer chain. In Figure 3.5 we can see the length of the bonds of different decamer configurations calculated by DFT and DFTD method, especially we can see that the bond lengths are not dependent on the configuration of the decamer but only on the method which we chose. Calculation with added dispersion gives shorter bonds lengths. In these graphs we can see the alternation of the bonds of two lengths. The length around 152 pm corresponds to the single bond and the length around 136 pm corresponds to the double bond.

Figure 3.5: Bond lengths in the polymer backbone calculated by the DFT and the DFTD method.

In Figures 3.6 and 3.7 we show the bond angles along the polymer backbone calculated by DFT and DFTD method. We can see especially for the DFT method that in the middle of the chain we have a periodic structure, which is broken at both ends of the chain. This fact is nicely seen on the dihedral angles along the polymer backbone. However, we can see that the bond angle along the polymer backbone varies only slightly between the values 117-120 degrees.

In Figure 3.8 we have the dihedral angle along the polymer backbone.

We can see that the dihedral angle changes quite significantly. We can see that the periodic structure, which can be seen in the middle, is broken at

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Figure 3.6: Bond angles along the polymer backbone calculated by the DFT method.

Figure 3.7: Bond angles along the polymer backbone calculated by the DFTD method.

the ends of the chain. Also the dihedral angle in the middle of the molecules has only two values, namely 130 degrees and 165 degrees, with the exception of geometry of the tail-to-tail configuration calculated by the DFTD method.

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Figure 3.8: Dihedral angles along the polymer backbone calculated by the DFT and the DFTD methods.

From now on we will discuss only the head-to-tail configuration as the most stable configuration of the polymer. For the following calculations we also need to know the equilibrium geometries of the anion-radical and the cation-radical. These calculations were done by the DFTD method in the atomic basis 6-31G*. This basis lacks the diffuse functions which are often added during the calculations of ion-radicals. The large size of our system with respect to the available computation resources was our reason for omit- ting them.

We show the difference between the calculated geometry of the neutral molecule and of the ion-radicals in the following figures. If we compare the geometry of the neutral chain and with that of the anion-radical and the cation-radical, respectively, we find out that the geometry has been only slightly changed. The motif of the helix is preserved. The relevant variables for comparing the shape of the polymer backbone are summarized in Figures 3.9-3.11. There are only slight changes in bond angles and dihedral angles along the polymer backbone. The lengths of the bonds in the polymer backbone (see Figure 3.11) changes more significantly. The difference in the bond lengths is largest in the middle of the polymer backbone, which is the place

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of the largest change in the spin density as we will show later.

Figure 3.9: Bond angles along the polymer backbone calculated by the DFTD method for the head-to-tail configuration.

Figure 3.10: Dihedral angles along the polymer backbone calculated by the DFTD method for the head-to-tail configuration.

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Figure 3.11: Bond lengths of the polymer backbone calculated by the DFTD method for the head-to-tail configuration.

In the modified Arkhipov model we expect that the charge transfer state can be represented as a hole located on the chain, which was excited by an absorbed photon, and an electron localized on one of the adjacent polymer chains. It was presumed that the electron is localized on the side group and the hole is trapped and delocalized on the main polymer backbone in a potential well caused by Coulomb interaction between the electron and the hole. We show that this notion is reasonable to certain degree.

In the following figures we present Mulliken spin and Mulliken charge densities. The purpose of this analysis was to estimate where the hole and the electron are localized on the cation-radical and on the anion-radical, respectively.

In Figure 3.12 we show Mulliken charge densities on the carbon atoms of the polymer backbone for the neutral molecule, the anion-radical and the cation-radical. These densities vary along the polymer chain only negligi- bly. From this picture one could wrongly assume that not the hole nor the electron are localized on the polymer backbone. We explain this discrepancy below.

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In Figure 3.13 we have summed Mulliken charge distributions on the side groups along the polymer backbone. We can see that there is a large difference in these densities between the neutral molecule and ion-radicals. This approach can give us a rough estimate of the localization of the charges forming the electric field that affects dissociation of the CT state. This approach, however, does not give us a right interpretation where the hole is localized in the cation-radical. To answer this question we need to look at the Mulliken spin densities. The reason, why knowing only the charge densities are not sufficient, is that the charge densities are shifted by the interaction of the electron spins, in open shell system compared to the situation in the closed shell system. The interaction between the same spins is repulsive.

Figure 3.12: Mulliken charge distributions along the polymer backbone calculated by the DFTD method for the neutral molecule, anion radical and cation radical, respectively.

In Figure 3.14 Mulliken spin densities on carbon atoms of the polymer backbone for the neutral molecule, the anion-radical and the cation-radical are shown. We can see, in contrast to the Mulliken charge densities, that these densities vary along the chain significantly. If we simply summed the spin densities on the polymer backbone we would see that for the anion radical 65 percent of the spin density is localized on the polymer backbone and

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Figure 3.13: Summed Mulliken charge distributions on each side group along the polymer backbone calculated by the DFTD method for the neutral molecule, anion radical and cation radical, respectively.

for the cation-radical even 69 percent of the spin density occurs on the polymer backbone. In Figure 3.13 we have summed Mulliken spin distributions on each monomer unit along the polymer backbone. We can see that the spin densities are distributed for both the cation-radical and the anion-radical in a similar way. When we look at the spin densities summed on each monomer unit we see that the hole for the case of the cation-radical and the electron for the case of anion-radical are highly delocalized. If we considered longer chain in our calculation we would obtain even more extended delocalization.

It was interesting to see the oscillation of the spin density along the carbon atoms in the main backbone. We compared these results to the results of the spin density calculations on the acetylene-decamer to show that it is not the effect of sidegroups. We calculated these spin densities in anion-radical and cation-radical of the acetylene-decamer in two geometries:

a) ideal conjugation, i.e. planar model of the trans-oligoacetylene, and b) in the geometry obtained for the main backbone of our decamer.

For the acetylene-decamer calculated with the geometry of the main backbone of our polymer we obtained essentially the same values as for our poly-

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Figure 3.14: Mulliken spin distributions along the polymer backbone calculated by the DFTD method for the neutral molecule, anion radical and cation radical, respectively.

mer. For the planar model we obtained Mulliken spin densities presented in Figure 3.16. These Mulliken spin densities are more symmetric with respect to the center of the chain, but otherwise the results are quite similar. It can be shown that the delocalization for the planar model is more extended as it has been expected.

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Figure 3.15: Summed Mulliken spin distributions on each monomer unit along the polymer backbone calculated by the DFTD method for the neutral molecule, anion radical and cation radical, respectively.

Figure 3.16: Mulliken spin distributions on carbon atoms along the planar acetylene calculated by the DFTD method.

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3.2 Mutual position of two chains

Although there are more accurate approaches to the problem the solution of mutual position of two molecules we stayed only with the most basic one based on the Van der Waals radii of atoms. We assumed that our molecules are rigid bodies and each of their atoms is enclosed in a sphere with the radius equal to the Van der Waals radius of the respective atom. The approaching molecules can at most touch by the spheres. The Van der Waals radii were taken from the database of physical constants in the Mathematica software [21]. They are shown in Table 3.3.

Table 3.3: Van der Waals radii of atoms Atom Van der Waals radius

Hydrogen 120 pm

Carbon 170 pm

Silicon 210 pm

As the next step we searched for the minimal distance between the cation- radical and the anion-radical. The search for the minimal distance was pro- grammed in the Mathematica software [21]. We did it in the following way.

We took both molecules (cation-radical and the anion-radical) in the standard orientation of the Gaussian software [16]. Next we introduced a virtual axis passing through the middle of the polymer, around which the polymer backbone is twisted. This axis was chosen in such a way that it coincides with the Cartesian axis x in the standard orientation of the molecule in the Gaussian software [16]. We refer to this axis as a coordinate axis. As the final step we searched for the optimal position between the molecules which gave us the minimal distance between the coordinate axes of the molecules.

Minimizing this distance is reasonable, because the initial step of the exciton dissociation is the creation of a CT state with the hole and the electron located at the adjacent molecules. This step is governed by the rate of the electron tunneling which depends on the tunneling distance. In order to maximize this tunneling rate we need to minimize the distance which the electron has to overcome. In the previous chapter we showed that the hole and the electron are located on the main backbone of the cation-radical and

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anion-radical, respectively. This conclusion is based on the analysis of the Mulliken spin densities on both radicals. As a result we should minimize the distance between the main backbones. As a simplification of this procedure we decided to minimize the distance between the coordinate axes of the adjacent chains. Under the approximations mentioned above, our system of two molecules have six degrees of freedom, namely:

a) the distance between the molecules calculated as the distance between their coordinate axes,

b) the angle between the coordinate axes,

c) two rotations, each corresponding to the rotation of the molecule around its coordinate axis, and

d) two shifts, each corresponding to the shift of the molecule in the direction of the coordinate axis.

Table 3.4: The minimal distance between the axes for various angles The angle between The distance between

the coordinate axes the coordinate axes

0 degrees 1455 pm

90 degrees 1305 pm

180 degrees 1459 pm

270 degrees 1284 pm

Searching for the minimal distance dependent on 5 variables (other degrees of freedom) is too difficult and we used the following simplifications. We took the angle between the coordinate axes as a parameter and limiting this parameter we looked for a solution only to several angles, namely, 0 degrees, 90 degrees, 180 degrees and 270 degrees. By this approach we decreased the number of variables to 4, but this is still not sufficient simplification for the solution of the problem. Next we found the minimal distance on the grid of 32×32 combinations of the angles corresponding to the rotations of the molecules around their respective coordinate axes and we retained the shift degrees of freedom fixed. On the grid the angles were varied by π/16 in the whole interval. After this optimal configuration had been found, we searched for the minimal distance on the grid of 3×3×15×15 combinations of the parameters. In this step the rotation degrees of freedom could change by π/16

Charles University in Prague Faculty of Science